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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 31 "Extended Interval Newton
 Method" }}{PARA 257 "" 0 "" {TEXT 256 234 "written by  Grimmer, Marku
s, Department of Mathematics, University of Wuppertal, Germany, http:/
/www.math.uni-wuppertal.de/wrswt \n<\251 1999-2002 Scientific Computin
g/Software Engineering Research Group, University of Wuppertal, German
y>" }{TEXT 257 1 "\n" }}{PARA 258 "" 0 "" {TEXT -1 63 "NOTE: This work
sheet demonstrates the use of the Maple package " }{TEXT 258 12 "intpa
kX v1.0" }{TEXT -1 25 " for interval arithmetic." }}}{EXCHG {PARA 259 
"" 0 "" {TEXT -1 102 "This document is not the package.  It only shows
 how to work with the functions and types provided by " }{TEXT 262 12 
"intpakX v1.0" }{TEXT -1 156 ".  You must create the package in an emp
ty directory before loading the package ( i.e., /usr/maple/intpakX/lib
)  Once created, load the package as follows:\n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 62 "restart;\nlibname:=\"C:/mylib/interval\", libn
ame;\nwith(intpakX);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(libnameG6$Q
2C:/mylib/interval6\"Q=C:\\Program~Files\\Maple~8/libF'" }}{PARA 7 "" 
1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n
" }}{PARA 7 "" 1 "" {TEXT -1 61 "Warning, the assigned name midpoint n
ow has a global binding\n" }}{PARA 7 "" 1 "" {TEXT -1 85 "Warning, the
 protected names ilog10, max and min have been redefined and unprotect
ed\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ar%#&*G%$&**G%#&+G%#&-G%#&/G%
-&Convex_HullG%(&arccosG%(&arcsinG%(&arctanG%&&cabsG%&&caddG%&&cdivG%*
&cdiv_optG%'&cmultG%+&cmult_optG%%&cosG%&&coshG%&&csubG%%&expG%+&inter
sectG%*&intpowerG%$&lnG%%&sinG%&&sinhG%%&sqrG%&&sqrtG%%&tanG%&&tanhG%'
&unionG%&EvalfG%6Interval_IntegerpowerG%4Interval_Round_DownG%2Interva
l_Round_UpG%-Interval_addG%0Interval_arccosG%0Interval_arcsinG%0Interv
al_arctanG%-Interval_cosG%.Interval_coshG%0Interval_divideG%-Interval_
expG%0Interval_hyp_rdG%0Interval_hyp_ruG%3Interval_intersectG%,Interva
l_lnG%2Interval_midpointG%5Interval_option_zeroG%/Interval_powerG%6Int
erval_range_valuesG%4Interval_reciprocalG%/Interval_scaleG%-Interval_s
inG%.Interval_sinhG%-Interval_sqrG%.Interval_sqrtG%2Interval_subtractG
%-Interval_tanG%.Interval_tanhG%/Interval_timesG%1Interval_trig_rdG%1I
nterval_trig_ruG%-Interval_ulpG%/Interval_unionG%/Interval_widthG%,add
infinityG%2centred_form_evalG%%cexpG%,cni_range3dG%2complex_disc_plotG
%5complex_polynom_plotG%2compute_all_zerosG%<compute_all_zeros_with_pl
otG%7compute_combined_rangeG%9compute_mean_value_rangeG%8compute_monot
onic_rangeG%=compute_naive_interval_rangeG%.compute_rangeG%0compute_ra
nge3dG%:compute_taylor_form_rangeG%*constructG%,expinfinityG%,ext_int_
divG%1horner_eval_centG%0horner_eval_optG%'ilog10G%+infinitylnG%%initG
%3interval_list_plotG%5interval_list_plot3dG%0intpakX_greaterG%/intpak
X_ilog10G%,intpakX_maxG%,intpakX_minG%$invG%&is_inG%$maxG%.max_abs_err
orG%$midG%)midpointG%$minG%'newtonG%,newton_plotG%1newton_with_plotG%.
powerinfinityG%#rdG%)rel_diamG%#ruG%,sqrinfinityG%-sqrtinfinityG%3subd
ivide_adaptiveG%1subdivide_equi3dG%6subdivide_equidistantG%1subtractin
finityG%.timesinfinityG%$ulpG%&widthG%)x0_startG" }}}{SECT 0 {PARA 3 "
" 0 "" {TEXT -1 31 "Extended Interval Newton Method" }}{PARA 0 "" 0 "
" {TEXT 259 13 "intpakX v1.0 " }{TEXT -1 234 "offers the Interval Newt
on Method for the computation/enclosure of all zeros of a continuously
 differentiable real function.\nYou can do the computation with or wit
hout graphical output of the intervals computed in the iteration steps
." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f:=x->sin(exp(sqrt(x-2)
));\nX:=[8.,10.];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6
\"6$%)operatorG%&arrowGF(-%$sinG6#-%$expG6#-%%sqrtG6#,&9$\"\"\"\"\"#!
\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG7$$\"\")\"\"!$\"#5
F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "compute_all_zeros(f,X
,0.001);" }}{PARA 6 "" 1 "" {TEXT -1 15 "`Digits = `, 10" }}{PARA 6 "
" 1 "" {TEXT -1 9 "`       `" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+
D/W%e*!\"*$\"+0B5!f*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;contains~e
xactly~one~zero.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+P7x0%)!\"*$
\"+,%*H1%)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;contains~exactly~one
~zero.G" }}{PARA 6 "" 1 "" {TEXT -1 9 "`       `" }}{PARA 6 "" 1 "" 
{TEXT -1 36 "`Number of enclosures of zeros: `, 2" }}{PARA 6 "" 1 "" 
{TEXT -1 33 "`Number of Iterations steps: `, 5" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 49 "compute_all_zeros_with_plot(f, X, 0.001, 10, 1
0);" }}{PARA 6 "" 1 "" {TEXT -1 15 "`Digits = `, 10" }}{PARA 6 "" 1 "
" {TEXT -1 8 "`      `" }}{PARA 6 "" 1 "" {TEXT -1 20 "`Iteration step
 `, 1" }}{PARA 6 "" 1 "" {TEXT -1 18 "`xold=`, [8., 10.]" }}{PARA 6 "
" 1 "" {TEXT -1 28 "`xnew1=`, [9.289288473, 10.]" }}{PARA 6 "" 1 "" 
{TEXT -1 27 "`xnew2=`, [8., 8.710711527]" }}{PARA 13 "" 1 "" 
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$!+,[:2<Fgeo7$Fggo$!+IF[2=Fgeo7$F\\ho$!+e1\"y!>Fgeo7$Faho$!+(eQ\"3?Fge
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$!+IP27KFgeo7$F`\\p$!+f;S7LFgeo7$Fe\\p$!+(eHFT$Fgeo7$Fj\\p$!+;v08NFgeo
7$F_]p$!+XaQ8OFgeo7$Fd]p$!+tLr8PFgeo7$Fi]p$!+-8/9QFgeo7$F^^p$!+I#pV\"R
Fgeo7$Fc^p$!+frp9SFgeo7$Fh^p$!+)3D]6%Fgeo7$F\\_p$!+;IN:UFgeo7$Fa_p$!+X
4o:VFgeo7$Ff_p$!+t)3gT%Fgeo7$F[`p$!++oL;XFgeo7$F``p$!+KZm;YFgeoFc`pFg`
pF[ap-%%FONTG6%%&TIMESG%'ITALICGFb_o-%*AXESTICKSG6%\"\"&%(DEFAULTG-F\\
[q6%F^[q%&ROMANGFb_o-%*AXESSTYLEG6#%&FRAMEG-%+AXESLABELSG6%Q\"xFg]oQ!F
g]o-F\\[q6#Fd[q-%(SCALINGG6#%.UNCONSTRAINEDG-%%VIEWG6$;$\"+e\"QeR)Fjjn
$\"+:J-i%)Fjjn;$!+&4ANa#F`bo$\"+eE?*Q\"Fjao" 1 2 0 1 10 0 2 9 1 3 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+D/W%e*!\"*$\"+0B5!f*F&" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%<~contains~exactly~one~zero.G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+P7x0%)!\"*$\"+,%*H1%)F&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%<~contains~exactly~one~zero.G" }}
{PARA 6 "" 1 "" {TEXT -1 11 "`         `" }}{PARA 6 "" 1 "" {TEXT -1 
36 "`Number of enclosures of zeros: `, 2" }}{PARA 6 "" 1 "" {TEXT -1 
32 "`Number of iteration steps: `, 5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 225 "\nOptionally, you can specify values for the number of Digits \+
used and for the number of iteration steps to be done\n(the latter for
 the graphical version only; it's annoying to get heaps of graphics wh
en you didn't want them.)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "\nTh
e enclosing intervals are stored in the global variable " }{TEXT 260 
6 "zeros." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "zeros[1]; zero
s[2];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+D/W%e*!\"*$\"+0B5!f*F
&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+P7x0%)!\"*$\"+,%*H1%)F&" }}
}}{EXCHG {PARA 261 "" 0 "" {TEXT 261 11 "Disclaimer:" }{TEXT -1 240 " \+
While every effort has been made to validate the solutions in this wor
ksheet, Waterloo Maple Inc. and the contributors are not responsible f
or any errors contained and are not liable for any damages resulting f
rom the use of this material." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 
1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }